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Upper bounds for the formula size of symmetric Boolean functions

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Abstract

We prove that the complexity of the implementation of the counting function of n Boolean variables by binary formulas is at most n 3.03, and it is at most n 4.47 for DeMorgan formulas. Hence, the same bounds are valid for the formula size of any threshold symmetric function of n variables, particularly, for the majority function. The following bounds are proved for the formula size of any symmetric Boolean function of n variables: n 3.04 for binary formulas and n 4.48 for DeMorgan ones. The proof is based on the modular arithmetic.

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Authors and Affiliations

  1. Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russia

    I. S. Sergeev

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  1. I. S. Sergeev
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Correspondence to I. S. Sergeev.

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Original Russian Text © I.S. Sergeev, 2014, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 5, pp. 38–52.

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Sergeev, I.S. Upper bounds for the formula size of symmetric Boolean functions. Russ Math. 58, 30–42 (2014). https://doi.org/10.3103/S1066369X14050041

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  • DOI: https://doi.org/10.3103/S1066369X14050041

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